Figure REF  shows the Fourier transform of the time series of the radial velocity at a fixed position (\(r=0.59\) ) in the equatorial plane for various \(Ra\)  at \(E=10^{-6}\)  and \(=0.001\) . For \(Ra=1.5\times 10^{7}\approx 1.4 Ra_c\) , where \(Ra_c\)  is the critical Rayleigh number for the onset of convection, the frequency spectrum shows a pronounced peak at \(\omega /\Omega =0.25\) , which corresponds to an oscillatory convective mode at low \(\)  [1]}. In figure REF  (a), there is also an obvious peak at \(\omega /\Omega \approx 0.5\) , twice the frequency of the primary mode. Similar doubling in frequency has been observed in recent numerical simulations [2]} and laboratory experiments [3]} using liquid metal in a rotating cylinder . As we shall show later, the doubling in frequency is accompanied by a doubling in the azimuthal wavenumber as well, which may result from the nonlinear self-interaction of the primary mode The nonlinear term of a flow \(u \propto \mathrm {e}^{\mathrm {i} (\omega t+m \phi )}\)  potentially has two components: \(m_0=0,\,\omega _0=0\)  and \(m_2=2m,\,\omega _2=2\omega \). For \(Ra=1.7\times 10^{7}\approx 1.6Ra_c\) , apart from similar peaks observed in figure REF (a), the spectrum exhibits two distinct peaks with similar amplitudes at \(\omega /\Omega =0.09\)  and 0.16, respectively ( figure REF (b)). These two peaks and the primary peak at \(\omega /\Omega =0.25\)  satisfy the triadic resonance condition, i.e. \(\omega _1 \pm \omega _2=\omega _0\) , suggesting a possible resonant triad between the primary convective mode and a pair of free modes.
